Electrostatic Resonance Oscillations of a Nonuniform Hot Plasma in an External Field

The frequency spectrum of a hydrodynamic model of a finite, warm, nonuniform plasma in an arbitrary external electric or magnetic field is considered. We find that the spectrum is real and the system stable, for an arbitrary configuration. A variational principle is given for estimating the eigenfrequencies. First-order perturbation theory is applied to a cylindrical plasma, and formulas are obtained for the first-order correction to the eigenfrequencies (resonances) for the case of an applied magnetic field or transverse electric field, arbitrary electron density $n_{00}\mathrm{}\left(r\right)\mathrm{}$, and arbitrary angular dependence $e^{i\mu \theta }$ ($\mu =0, \pm 1, \pm 2, \cdots$), the effect of the applied fields on the zero-order electron density being included. We find that for $\mu \ne 0$, the modes have a twofold degeneracy, and that a uniform axial magnetic field splits the resonances in two. The first-order correction to the resonances is found to vanish for a uniform transverse electric or magnetic field. These results are discussed relative to other models and to experiment, and appear to be in agreement with the available experimental data for the behavior of the main dipole resonance in both transverse and axial magnetic fields.